Active Suspension System’s Non linearity on The ... · slow-active suspension system and...

Paper: ASAT-16-083-ST

16th

International Conference on

AEROSPACE SCIENCES & AVIATION TECHNOLOGY,

ASAT - 16 – May 26 - 28, 2015, E-Mail: [email protected]

Military Technical College, Kobry Elkobbah, Cairo, Egypt

Tel : +(202) 24025292 – 24036138, Fax: +(202) 22621908

Investigation of Slow-Active Suspension System’s Non-linearity on

The Performance of The Anti-lock Brake System

A. M. Onsy*, A. M. Sharaf

†, M. M. Ashry

‡ and S. M. El-demerdash

§

Abstract: This paper presents an advanced integrated controller design for military vehicle

slow-active suspension system and anti-lock brake system (ABS) considering the effect of the

suspension components non-linearity on the braking dynamics. The presented investigation is

applied to a quarter vehicle model using the Magic Formula tire model and the actual

characteristics for suspension damper (throttle valve) and spring (gas spring). The

suspension controller is designed based on LQR using a linearized vehicle model and the

ABS controller is designed based on ON/OF control methodology using tire skid ratio of 15%

as a reference value. The mathematical derivations as well as the control techniques are

numerically implemented in MATLAB environment. Several simulation results are carried

out considering typical performance measures such as stopping time, stopping distance and

vehicle body vertical acceleration. The results illustrated that, the integrated controller reduces

the stopping distance on different road surfaces and therefore improves the braking

performance to avoid or reduce the impact of vehicle accident.

Keywords: Slow-active suspension, Anti-lock braking system, Vehicle dynamics and control.

Nomenclatures

fA Aerodynamic reference area[m2]

B Magic formula stiffness factor [-]

dC Vehicle aerodynamic drag coefficient [-]

dF Aerodynamic drag force [N]

sF Passive suspension forces [N]

tF Tire dynamic forces [N]

xF Tire longitudinal friction forces [N]

rF Tire rolling resistance forces [N]

zF Tire normal load [N] g Gravity acceleration [m/s2]

wJ Unsprung mass moment of inertia [kg.m2]

tK Tire stiffness coefficient [N/m]

bm, w

m Vehicle body and wheel masses respectively [kg]

* PhD Student, Automotive Engineering Dept., Military Technical College

† Associate Professor, Head of Automotive Engineering Dept., Military Technical College

‡ Associate Professor, Optoelectronics Dept., Military Technical College, Cairo, Egypt

§ Professor, Helwan University, Faculty of Engineering and Technology, Mataria, Cairo, Egypt


aP ,dP Actual and demand brake pressures respectively [N/m

2]

wR Tire dynamic radius [m]

V Vehicle forward speed [m/s]

bx Vehicle braking deceleration [m/s2]

oz Road displacement [m]

bz , wz Sprung mass and unsprung mass vertical displacements [m]

b Disk brake coefficient of friction [-]

1. Introduction

The military vehicles are regularly working on different terrains and thus needs high stability

and safety during all operating conditions [1, 2]. The active suspension systems and anti-lock

brake systems (ABS) are the vehicle chassis controllable systems which affect vehicle

dynamics, stability and safety. Much research has been carried out to design different active

suspensions as well as ABS systems using different control theories to improve the ride

comfort and braking performance [2-12]. The controllers of these chassis controllable systems

are designed based on an individual manner ignoring their interaction in practice. Recently

[13-15], more attention is given to design an integrated controller for the so called slow-active

suspension and ABS systems considering the dynamic coupling between them through the

tire/road interaction forces.

The motivation of the presented research arises from the fact that, the tire forces in

longitudinal direction - either due to traction or braking - are fundamentally regulated by the

available adhesion potential at the tire-ground contact patch. This potential is dependent upon

the road friction (μ) and the tire imposed vertical force. Sequentially, this tire vertical force is

predominantly restricted by its vertical oscillation which is highly controlled by the properties

of the suspension system. The presented paper exploits the aforementioned philosophy in

favor of both ride and braking dynamics.

The details of the modelling basis are reported by the authors in [14] including an integrated

control system of anti-lock brake system (ABS) and linearized slow-active suspension system.

A linear optimization technique was followed to improve the ride dynamics in terms of

suspension working space, body vertical acceleration and tire dynamic normal load.

In this paper, the reported initial approach is extended by inclusion of non-linear slow-active

controlled suspension system characteristics. Additionally, the controller is designed based on

the optimal control theory; taking into account changes in vehicle speed during the braking

process. The suspension controller feedback gains are selected to minimize variations in tire-

road normal forces. The system optimization based on the linear optimal control theory is

carried out only for the linear system and the calculated feedback gains are used in both linear

and non-linear cases. The ABS controller is designed to control the skid ratio by a

proportional directional valve with a time delay of 0.1 sec.


2. Non-Linear Suspension System Model

2.1 Equations of Motion:

A quarter vehicle model is employed to correlate the vehicle ride dynamics due to vertical

oscillations and the vehicle performance in the longitudinal direction due to the application of

brakes as depicted in Fig. 1. The model consists of an unsprung or wheel mass wm , which is

connected to approximately a quarter of the vehicle’s sprung mass bm by a single acting

hydraulic cylinder.

The hydro-pneumatic suspension system consists of gas spring which is equivalent to

mechanical spring and flow control valve (throttle valve) which is equivalent to damper. The

ride model possesses two degrees of freedom in vertical direction namely; the vertical

displacement of the vehicle quarter body sprung mass bz and the vertical displacement of

the wheel center wz . Considering the Newton second law, the equations of motion are

written as follows:

w w t O w S

Suspension ForceTire Force

m z k z z F (1)

b b Sm z F (2)

With the hydro-pneumatic based systems, active control is exercised by regulating the volume

of oil which flows into or out of the strut. This system uses a low bandwidth proportional

directional control valve. The fluid pressure is generated by a hydraulic pump with

accumulators providing transient flow requirements above the average. For a slow-active

suspension system, the suspension force SF is calculated as follows:

'

S st sF A P (3)

Where stA is the area of the piston strut given by: 2

st stA R , stR is the strut piston

radius. The strut perturbation pressure 'sP is equal to gas spring perturbation pressure 'gP

in addition to the pressure drop across the throttle valve (damping element) dP .

' '

s g dP P P (4)

Additionally, the dependency of the pressure drop across the throttle valve dP on the flow

rate through the throttle valve DQ considering different constants for throttle valve 1C are

depicted in Fig 2. and can be mathematically represented as follows: 2

1d DP C Q (5)

Assuming an incompressible fluid flow, the rate of change of the fluid volume DQ can be

expressed in terms of suspension relative velocity between the vehicle body and wheel

w bz z :

D st w bQ A z z (6)


wm

Pro

po

rtio

na

l

Dir

ec

tio

na

l

Co

ntr

ol V

alv

eHydraulic

Pump

Oil Tank

Gas

Spring

Ac

cu

mu

lato

r

for

no

ise

att

en

ua

tio

n bm

zFxF rF

wz

bz

oz

bT tK

r

bV x

Direction of motion

b b dm x F

bm g

z

Throttle

Valve

Sin

gle

- A

cti

ng

Hy

dra

ulic

Cy

lin

de

r

sF

x

Fig. 1.Representation of the Quarter Vehicle Model of the IntegratedSystem

2.2The Flow Control Valve

The non-linear characteristics of the flow control or throttle valve used in this paper is based

on a typical valve type DV20-1-10/M which is produced by REXROTH Ltd [6]. The valve

characteristics for different throttling positions is shown in Fig 2.

Fig. 2.Non-linear Characteristics of the Flow Control Throttle Valve

0 25 50 75 100 125 150 175 200 225 250 275 3000

50

100

150

200

250

300

Flow Rate (L/min)

Pre

ssure

Dro

p (bar)

C1=1.427E12

C1=7.84E11

C1=1.038E14

C1=1.98E13

C1=4.165E12

Cs=2000 N.S/m

C1=3.97E11


2.3The Gas Spring

The gas spring static pressure gstP and gas spring static volume gstV are set according the

quarter body mass bm to be supported and suspension stiffness suspK and are given by:

bgst

st

m gP

A

(7)

2

st gst

gst

susp

A PV

K

(8)

Due to the large variation in the strut pressure from static equilibrium value, the gas spring,

throttle valve and proportional directional control valve can no longer be considered as linear,

see Fig 3. To incorporate the non-linearity of these components their equations require some

small changes. The change of the pressure and volume of gas depend on the volume of the

fluid that displaced into, or withdraw from the gas spring according to adiabatic gas law as

follows:

'

' gst

g gst

gst g

VP P

V V

(9)

Considering equations (5) and (9), equation (4) can be rewritten as follows:

2

'

'

1

gst

s gst D

gst g

VP P C Q

V V

(10)

Where '

gV is the volume compressed by the fluid displaced into the gas spring and is given

by: '

g fluidV V (11)

fluid st W b AV A z z V (12)

Where AV is fluid volume, which passes into the suspension system through the valve.

Fig. 3. Non-linear Characteristics for the Gas Spring [6]

0.16 0.2 0.24 0.28 0.32 0.360

2

4

6

8

10

12

Gas Volume (Litre)

Gas P

ressure

(M

pa)

-0.1 -0.05 0 0.05 0.10

2

4

6

8

10

12

Displaced Gas Volume (Litre)

Gas P

ressure

(M

pa)


The pressure feedback signal pe is modified with respect to adiabatic pressure change as

follow

' (

gst gst gst gst

p

gst st W bgst g

P V P Ve

V A z zV V

(13)

The error is given by the following equation:

gdf PfP P PfD Perror P K e K e [6]

(14)

gst

P A

gst

Pe Q

V

(15)

2.4 Proportional Directional Control Valve

The control valve is modeled using a standard relationship in which the flow rate ADQ is

proportional to the pressure drop VP across the valve from supply to strut or from strut to

atmosphere. Therefore, the valve supply voltage '

VV is given as follow:

'

V PAV K error (16)

Accordingly, the demand flow rate ADQ is given as follows:

0.1 PVAD PA V

V

KQ K error P

R

[6]

(17)

VP , is given in bar as follows:

................. 0

................. 0

susp g

V

g atm

P P if errorP

P P if error

(18)

The supply pressure used in this work 100 suspP bar and according to the valve

characteristics the maximum flow rate max 25 / minAQ L and

10 32.664 10 / sec/PVK m Amp .

VA is calculated according to the flow diagram shown in Fig. 4 and then used in equation (12).

It is a complicated task to calculate the feedback and feed/forward gains using the linear

optimal control theory for this non-linear system. Therefore the system optimization based on

the linear optimal control theory is carried out only for the linear system and the calculated

feedback and feed/forward gains are used in both linear and non-linear cases.


PAK 1 RPVK

PD Controller

S

Err

or

VV I ADQAQ2

2 22

V

V V VS S

K2 2

2

2 d d d

d

S S

b wZ Z

bZ

gP

DK

PK

go

st

go

PA

V

2 nd Order Low Pass Filter

Demand gas

pressure signal

gP

pe

gdPgdfP

proportional control valve

Suspension System

State Estimator

Suspension

System

Fig. 4.Block-Diagram for the Slow-active Suspension System Controller

3. Anti-lock Brake System Controller Design

The detailed modelling work of the Anti-lock Brake System is presented by the authors in

[14], for the purpose of completeness of this paper; the modelling work can be summarized as

follows:

3.1 The Longitudinal Dynamics

Referring to Fig. 1, the tire vertical load zF is simply calculated as follow:

z b w b b w w

Tire Vertical Load Tire Dynamic LoadTire Static Load

F m m g m z m z (19)

Also the following equations are derived:

21 1

2 b x d b

fb w

x F C A xm m

(20)

1

x r w b

w

F F TJ

R (21)

The Pacejka's Magic Formula is used to estimate the barking force xF at the tire-ground

contact patch, more details can be found in [14]. The formula relates the percentage of wheel

skid ratio S and tire vertical force zF as follows:

1 100 %w

b

RS

x

(22)

1 1sin tan tanx m m m m m mF D C B S E B S B S

(23)


Where: mC is a constant, , ,m m mB D E are functions of the tire normal load zF and slip

[14].

The tire braking force versus skid ratio is simulated for the model with and without

suspension as shown in Fig. 5. It is obvious that, the potential of fluctuating tire dynamic load

and braking force accordingly is increased when the suspension system features are

considered. This result conforms to the well-known fact that, the design of suspension system

is highly affect the dynamics of forces at the tire ground contact area in both vertical and

longitudinal directions. This effect should be carefully considered whenever the stability of

the vehicle during braking maneuver is a prime of interest. On the other hand, it is expected

that the ride comfort will be significantly affected in an adverse manner.

Fig. 5.The Dependency of the Generated Braking Forces on the Suspension System

3.2 The ABS Controller

The proposed control strategy combines a separate controller for the anti-lock brake system

and a non-linear slow-active suspension system with a governing algorithm to coordinate the

two controllers. Fig. 6 shows schematic drawings for the integrated control system comprising

a slow-active controlled suspension and controlled ABS brake system.

The fundamental function of the ABS control system is to regulate the friction utilization at

the tire-ground contact area during braking process [11, 12]. Accordingly, it prevents the tire

from excessive skid particularly at higher braking forces. The tire skid ratio is limited by a

desired value of 15% to maintain the peak braking force over different levels of road adhesion

and tire vertical load. The estimated skid ratio is then compared with the value of the desired

skid ratio to produce an error signal. For each value of the error, the controller returns 1 if the

value is greater than zero and hence the pressure is increased, returns 0 if the error equals zero

and hence the pressure remains unchanged, and -1 if it is less than zero, and hence the

pressure is reduced. On the other hand, according to the signal of the vehicle speed, the

suspension controller modifies the feedback gains to minimize the fluctuations of tire normal

force. The angular speed of the tire, deceleration, vehicle speed and hydraulic pressure of the

brake fluid are measured from several sensors in order to estimate the tire skid ratio.

According to the demand hydraulic pressure dP , a proportional directional control valve is

employed to modulate the hydraulic brake pressure aP as shown in Fig. 6. The directional

control valve is modeled as a first order differential equation with a time delay t as follows:

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

4000

5000

6000

Time (Sec)

Bra

ke F

orc

e (N

)

without suspension

with suspension


1 1a a dP P P

t t

(23)

wm

Pro

po

rtio

na

l

Dir

ec

tio

na

l

Co

ntr

ol V

alv

e

Hydraulic

Pump

Oil Tank

Suspension

Controller

Gas

Spring

Ac

cu

mu

lato

r

for

no

ise

att

en

ua

tio

n bm

Brake

System

Error > 0

Error = 0Error

Yes

No

Yesd bP P

d bP P P

d bP P P

1

v

1

wR

w

vActual Skid

Ratio

0

-1

+1

Desired Skid

Ratio (15%)

No

wz

bz

oz

b

Vx

Direction

of motion

Ve

hic

le S

pe

ed

bw

zz

bzgP

z

Throttle

Valve

Sin

gle

- A

cti

ng

Hy

dra

ulic

Cy

lin

de

r

ABS

Actuator

ABS Controller

1- Accelerometer

2- Pressure Transducer

3- Angular Speed Sensor

4- Displacement Transducer

1

2

3

4

r

x

Fig. 6.Block Diagram for the ABS and Non-linear Slow-active Suspension Controllers

The resulting brake torque bT can be calculated as is a linear function of the brake pressure

aP , the area of the wheel cylinder wcA , the disk brake lining friction coefficient b , the

effective disk brake radius r and the number of friction surface sn as follows:

b a wc b s

Piston Force

T P A n r (24)


4. Road Profile

It is widely recognized that the road surfaces approximate to Gaussian random processes,

having a power spectral density (PSD) of the form:

1n

c

n

R VPSD f

f

(25)

cR is the road roughness coefficient (see Table 1), V is the vehicle speed (m/sec), f is the road

excitation frequency (Hz), n is an exponent with a recommended value of 2.5.The road input

is represented as linear filtered white noise process by the following first order differential

equation in time domain:

2o W oz F z B w t (26)

Where: 2

2 , 1W

F V B , is the cut-off frequency wave number =0.005 (cycle/m),

V is the vehicle speed [6].

5. Results and Analysis

The derived integrated control algorithm as well as the proposed optimization technique is

implemented using the MATLAB package. For the purpose of performance comparison,

several simulation results have been reported incorporating Anti-Lock Braking system with

passive suspension system and non-linear slow-active suspension system. For the braking

performance evaluation, several measures are used such as the vehicle forward speed, the

wheel rotational speed, the braking pressure, the tire skid ratio and finally the stopping

distance. Other important results such as the effect of the road roughness coefficient and the

coefficient of adhesion on the braking performance are also considered for both systems. In

carrying out these calculations, the numerical values of the vehicle main parameters are used

and given in Appendix (I).

The results of simulation are illustrated in Figs. 7-13. The initial speed of the vehicle is 100

Km/h (27.8 m/s) and the brake pedal is applied while driving on a wet road with minor

roughness coefficient 850 10cR .

Primarily, Power spectral density results of body acceleration and DTL for the non-linear

passive, linear slow-active and non-linear slow-active system are shown in Figs. 7-8, around

the body resonance peak, the linear and non-linear systems behave very similarly to each

other. The only slight difference appears to be above 9Hz and this is almost certainly due to

the different levels of effective damping between a linear value of 2000 Ns/m and throttle

valve position 2.5.

The result of implementing the integrated controller, ABS and non-linear slow-active

suspension system, reveals an improved stopping time as shown in Figs. 9. The integrated

controller monitors both vehicle and wheel rotational speed sensors at all times. Therefore, it

regulates the braking pressure to the wheel cylinder up and down, Fig. 11, according to the

desired skid ratio as shown in Fig. 10. As a result, the wheel is prevented to excessively lock-

up and thus decelerates at similar rate to the vehicle body. A comparison between the

integrated system and the different suspension systems with ABS are indicated in terms of the

stopping time and the stopping distance as shown in Table (1).


Table (1) Comparison between Different Suspension Systems Integrated with ABS system

Braking distance with Passive suspension system (m) 72

Braking distance with Linear slow-active suspension system (m) 68

Braking distance with Non-Linear slow- active suspension (m) 103

Braking distance with Non-Linear slow-active suspension 95

Fig. 7.PSD of Body Acceleration for Linear versus Nonlinear Slow-Active System

Fig. 8.PSD of Tire Dynamic Load for Linear versus Nonlinear Slow-Active System

To investigate the improvements afforded by the proposed system, Fig. 9illustrates the

outcomes of implementing the integrated control system in comparison to a conventional

ABS fitted with other suspended systems. The evaluation is carried out in terms of the

stopping distances for all systems, it could be seen that the braking distance of different

suspension systems is differs due to changing of dynamic tire load which lead to changing in

normal force and from the beginning of this chapter the braking force is changing linearly

with normal force and off course inversely with braking distance. The analysis of this figure

shows that the nonlinear-passive suspension system is the highest braking distance and linear

slow-active suspension system is the lowest one.

0 2 4 6 8 10 12 14 150

0.25

0.5

0.75

1

1.25

1.5

Frequency (HZ)PSD

(V

ert

ical A

ccele

rati

on) m

2/s4

Nonlinear-Passive

Linear-slow-active

Nonlinear-slow-active

0 2 4 6 8 10 12 14 150

200

400

600

800

1000

Frequency (HZ)

PSD

(T

ire D

ynam

ic L

oad) N

2/H

Z

Nonlinear-Passive

Linear-slow-active

Nonlinear-slow-active

Damping coefficient for linear Slow-Active System is 2000 N

s/m Throttle position for Non-linear Slow-Active System is 2.5.

Damping coefficient for linear Slow-Active System is 2000 N

s/m Throttle position for Non-linear Slow-Active System is 2.5.


Fig. 9.Improvement of Braking Performance (Vehicle Speed) using ABS System

Fig. 10.Improvement of Braking Performance (Skid Percentage) using ABS System

Fig. 11.Improvement of Braking Performance (Brake system pressure) using ABS System

0 1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

Time (sec)

Tir

e s

peed, V

ehic

le s

peed (m

/s)

Tire speed without ABS

Tire speed with ABS

Vehicle speed without ABS

Vehicle speed with ABS

0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

Time (sec)

Tir

e s

kid

rati

o (%

)

Without ABS

With ABS

0 1 2 3 4 5 6 7 8 9 100

0.25

0.5

0.75

1

1.25

1.5

1.75

2x 10

7

Time (sec)

Bra

ke p

ressure

(pa)

Without ABS

With ABS


Fig. 12.Improvement of Braking Performance (Braking Distance) using ABS System

Fig. 13.Comparison of Braking Distance considering Different types of Suspension Systems

6. Conclusions

A nonlinear mathematical formulation for a quarter vehicle model has been derived to study

the effect of non-linear components of slow-active suspension on both ride and braking

dynamics. The coupling between both systems dynamics is arises from the dynamic normal

load at the tire/terrain contact area. An integrated controller is designed to minimize the

dynamic tire load and maximize the maximum tire/terrain brake force. For this purpose, two

controllers, slow-active suspension controller and anti-lock braking system controller, are

combined together to the so called integrated controller. It has been illustrated that, the

integrated controller reduces the stopping distance on different terrain conditions considering

the suspension system components nonlinearities. And, therefore improves the braking

distance to avoid or reduce the impact of vehicle accident. The proposed controller is suitable

for enhancing the vehicle mobility during traction by controlling the tire/road interaction

forces and preventing the physical fatigue of the driver due to vehicle vibration.

0 1 2 3 4 5 6 7 8 9 100

25

50

75

100

125

150

Time (sec)

Bra

kin

g d

ista

nce (m

)

With ABS

Without ABS

0 1 2 3 4 5 6 7 80

20

40

60

80

100

120

140

Time (sec)

Bra

kin

g D

ista

nce (m

)

Passive Suspension

Linear Slow Active

Non-Linear Passive

Non-Linear slow-Active


References

[1] J. Y. Wong "Theory of ground vehicles", 4th

ed., Hoboken, N.J., Wiley, 2008.

[2] S. Sulaiman, P. M. Samin, H. Jamaluddin, R. Abd Rahman, and M. S. Burhaumudin,

"Ground hook Control of Semi-Active Suspension for Heavy Vehicle", International

Journal of Research in Engineering and Technology (IJRET), Vo.1, No. 3, 2012.

[3] G. Koch, and T. Kloiber "Driving State Adaptive Control of an Active Vehicle

Suspension system", IEEE Transactions on Control Systems Technology, Vol. 22, No. 1,

2014.

[4] C. Gohrle, A. Schindler, A. Wagner, and O. Sawodny "Design and Vehicle

Implementation of Preview Active Suspension Controllers", IEEE Transactions on

Control Systems Technology, Vol. 22, No. 3, 2014.

[5] B. Gao, J. Darling, DG. Tilly, RA. Williams, A. Bean, and J. Donahue "Control of a

Hydropneumatic active Suspension Based on a non-linear Quarter-Car Model",

Proceedings of the Institution of Mechanical Engineers, Part I-Journal of Systems and

Control Engineering", Vol. 220, pp.15-31, 2006.

[6] El-Demerdash, S.M. and Crolla, D.A. ''Effect of non-linear components on the

performance of a hydro-pneumatic slow-active suspension system", Proceedings of the

Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 1996.

[7] J. Fen, S. Zheng, and F. Yu "Bandwidth-Limited Active Suspension Controller for an

Off-Road Vehicle Based on Co-Simulation Technology", Mech. Eng. China, 3(1), pp.

111-117, 2008.

[8] M. Sibielak, J. Konieczny, J. Kowal, W. Raczka, and D. Marszalik, "Optimal Control of

Slow-Active Vehicle Suspension – Results of Experimental Data", Journal of Low

Frequency Noise Vibration and Active Control, Vol. 32, pp. 99-116, 2013.

[9] MM. ElMadany "Control and Evaluation of Slow-Active Suspension with Preview for a

Full Car", Mathematical Problems in Engineering, No. 375080, 2012.

[10] SF. Van der Westhuizen, and PS. Els "Slow Active Suspension Control for Rollover

Prevention", Journal of Terramechanics, Vol. 50, pp. 29-36, 2013.

[11] Nilanjan Patra, Kalyankumar Datta,"Sliding mode Controller for Wheel-slip Control of

Anti-lock Braking System", IEEE International Conference on Advanced

Communication Control and Computing Technologies (ICACCCT) 2012.

[12] Ayman A. Aly, El-Shafei Zeidan, Ahmed Hamed, Farhan Salem, "An Antilock-Braking

Systems (ABS) Control: A Technical Review", Intelligent Control and Automation,

2011, 2, 186-195 doi:10.4236/ica.2011.

[13] S.M. El-Demerdash, M.B. Abdelhady, M.A. AlNashar and A.S. Emam, "Effect of

Vehicle Suspension Control on the Performance of Anti-Lock Brake Systems", IMechE

International Conference on Braking Technology, St William's College, York, May 7-9,

2006, UK.


[14] A.M. Onsy, A.M. Sharaf, M.M. Ashry and S.M. Eldemerdash "Effect of Slow-Active

Suspension Controller Design on the Performance of Anti-lock Brake System", 15th

International Conference on ASAT- 15, Military Technical College, Kobry Elkobbah,

Cairo, Egypt, 2013.

[15] J. Song "Development and Comparison of Integrated Dynamics Control Systems with

Fuzzy Logic Control and Sliding Mode Control", Journal of Mechanical Science and

Technology, Vol. 27(6), pp. 1853-1861, 2013.

Appendix (I) Main parameters of the quarter vehicle model and the ABS

Parameter Symbol Value Unit

Quarter of the vehicle body or sprung mass bm 578 [kg]

Unsprung masses wm 58.25 [kg]

Engine and wheel moments of inertia wJ 13.7 [kgm2]

Passive spring stiffness coefficients sK 25 [kN/m]

Passive damping coefficients sC 1 [kNs/m]

Tire vertical stiffness coefficients tK 255 [kN/m]

Vehicle aerodynamic frontal area fA 2.04 [m

2]

Vehicle aerodynamic drag coefficient dC 0.539 [--]

Wheel cylinder piston radius pr 0.0185 [m]

Tire dynamic radius wR 0.273 [m]

Front and rear lining coefficient of friction b 0.35 [--]

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